Chapter 20
Testing Hypotheses
About Proportions
Example
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Census data for a certain county show that 19% of the
adult residents are Hispanics. Suppose 72 people are
called for jury duty and only 9 of them are Hispanic.
Does this apparent underrepresentation of Hispanics call
into question the fairness of the jury selection system?
A Trial as a Hypothesis Test
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Think about the logic of jury trials:
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To prove someone is guilty, we start by assuming they are
innocent.
We retain that hypothesis until the facts make it unlikely
beyond a reasonable doubt.
Then, and only then, we reject the hypothesis of innocence
and declare the person guilty.
The same logic is used in hypotheses testing:
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We begin by assuming that a hypothesis is true.
Next we consider whether the data are consistent with the
hypothesis.
If they are, all we can do is retain the hypothesis we started
with. If they are not, then like a jury, we ask whether they
are unlikely beyond a reasonable doubt.
Hypotheses
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Hypotheses are working models that we adopt temporarily.
Our starting hypothesis is called the null hypothesis, denoted
by H0, which specifies a population model parameter of
interest and proposes a value for that parameter.
 For example, H0: p = 0.19, as in the example.
We usually write down the null hypothesis in the form
H0: parameter = hypothesized value.
The alternative hypothesis, which we denote by HA, contains
the values of the parameter that we consider plausible when
we reject the null hypothesis.
We want to compare our data to what we would expect given
that H0 is true.
We then ask how likely it is to get results like we did if the null
hypothesis were true.
P-Values
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The statistical twist is that we can quantify our level
of doubt.
 We can use the model proposed by our
hypothesis to calculate the probability that the
observed statistic value occur if the null
hypothesis is correct.
 This probability is called a P-value.
 If the P-value is low enough, we’ll “reject the null
hypothesis,” since what we observed would be
very unlikely were the null model true.
What to Do with an “Innocent” Defendant
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If the evidence is not strong enough to reject the
presumption of innocent, the jury returns with a verdict of
“not guilty.”
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The jury does not say that the defendant is innocent.
All it says is that there is not enough evidence to convict,
to reject innocence.
The defendant may, in fact, be innocent, but the jury has
no way to be sure.
Said statistically, we will fail to reject the null hypothesis.
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We never declare the null hypothesis to be true, because
we simply do not know whether it’s true or not.
Sometimes in this case we say that the null hypothesis
has been retained.
The Reasoning of Hypothesis Testing
There are four basic parts to a hypothesis test:
1. Hypotheses
2. Model
3. Mechanics
4. Conclusion
The Reasoning of Hypothesis Testing (cont.)
1. Hypotheses
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The null hypothesis: To perform a hypothesis test, we
must first translate our question of interest into a
statement about model parameters.
H0: parameter = hypothesized value.
The alternative hypothesis: HA contains the values of the
parameter we consider plausible if we reject the null.
There are three possible alternative hypotheses:
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HA: parameter < hypothesized value
HA: parameter ≠ hypothesized value
HA: parameter > hypothesized value
The Reasoning of Hypothesis Testing (cont.)
2. Model
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To plan a statistical hypothesis test, specify the model
you will use to test the null hypothesis and the
parameter of interest.
All models require assumptions, so state the
assumptions and check any corresponding conditions.
Your plan should end with a statement like
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Because the conditions are satisfied, I can model
the sampling distribution of the proportion with a
Normal model.
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Watch out, though. It might be the case that your
model step ends with “Because the conditions are
not satisfied, I can’t proceed with the test.” If that’s
the case, stop and reconsider.
The Reasoning of Hypothesis Testing (cont.)
2. Model
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Each test we discuss in the book has a name that you should
include in your report.
The test about proportions is called a one-proportion z-test.
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The conditions for the one-proportion z-test are the same
as for the one proportion z-interval. We test the
hypothesis H0: p = p0
z
using the statistic
where
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SD  pˆ  
 pˆ 
p0 
SD  pˆ 
p0 q0
n
When the conditions are met and the null hypothesis is true,
this statistic follows the standard Normal model, so we can
use that model to obtain a P-value.
The Reasoning of Hypothesis Testing (cont.)
3. Mechanics
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We place the actual calculation of our test statistic from
the data. Different tests will have different formulas and
different test statistics.
Usually, the mechanics are handled by a statistics
program or calculator, but it’s good to know the formulas.
The ultimate goal of the calculation is to obtain a P-value.
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The P-value is the probability that the observed
statistic value could occur if the null model were
correct. If the P-value is small enough, we’ll reject the
null hypothesis.
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Note: The P-value is a conditional probability—it’s the
probability that the observed results could have
happened if the null hypothesis is true.
The Reasoning of Hypothesis Testing (cont.)
4. Conclusion
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The conclusion in a hypothesis test is always a
statement about the null hypothesis.
The conclusion must state either that we reject or
that we fail to reject the null hypothesis.
And, as always, the conclusion should be stated in
context.
Alternative Hypothesis
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HA: parameter ≠ value is known as a two-sided alternative
because we are equally interested in deviations on either
side of the null hypothesis value.
For two-sided alternatives, the P-value is the probability of
deviating in either direction from the null hypothesis value.
Alternative Hypothesis (cont.)
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The other two alternative hypotheses ( < or > ) are called
one-sided alternatives.
A one-sided alternative focuses on deviations from the
null hypothesis value in only one direction.
Thus, the P-value for one-sided alternatives is the
probability of deviating only in the direction of the
alternative away from the null hypothesis value.
P-Values and Decisions:
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How small should the P-value be in order for you to reject
the null hypothesis?
It turns out that our decision criterion is contextdependent.
 When we’re screening for a disease and want to be
sure we treat all those who are sick, we may be willing
to reject the null hypothesis of no disease with a fairly
large P-value.
 A longstanding hypothesis, believed by many to be
true, needs stronger evidence (and a correspondingly
small P-value) to reject it.
Another factor in choosing a P-value is the importance of
the issue being tested.
P-Values and Decisions (cont.)
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Your conclusion about any null hypothesis should be
accompanied by the P-value of the test.
 If possible, it should also include a confidence interval
for the parameter of interest.
Don’t just declare the null hypothesis rejected or not
rejected.
 Report the P-value to show the strength of the
evidence against the hypothesis.
 This will let each reader decide whether or not to reject
the null hypothesis.
Group Exercises
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An insurance company checks police record on 590
accidents selected at random and notes that teenagers
were at the wheel in 88 of them. Construct the 95%
confidence interval for the percentage of all auto
accidents that involve teenager drivers.
According to June 2004 Gallup poll, 28% of Americans
said there have been times in the last year when they
haven’t been able to afford medical care. Is this proportion
higher for black Americans than for all Americans? In a
random sample of 801 black Americans, 38% reported
that there had been times in the last year when they had
not afford medical care. Test an appropriate hypothesis
and state your conclusion.
Homework Assignment
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Page 527 - 530
Problem # 1, 3, 7, 21, 23, 25, 29.